Higgs Boson Machine Learning Challenge

The goal of this challenge is to identify which simulated ATLAS experiment proton-proton collision observations contain evidence of boson pairs (“signal”), and which do not (“background”).

The challenge is hosted by Kaggle and provided by The Atlas Experiment. Technical details were found in the Laboratoire de l’Accélerateur Linéaire technical documentation PDF. Attempts at summarizing these are my own, and may be imperfect (or completely off - I'm not a particle physicist!) Pull requests to correct these are welcome.

The Data

Provided were a training set of 250,000 observations and a test set of 500,000 observations. Unusable or unknown quantities are reported as -999.0.

The variables include 17 “primitive” quantities and 13 “derived” quantities. Also included is the EventId identifying the observation. The training set also includes the goal label (b or s) and a “Weight” metric not to be used in classification. For this reason I've stripped it out of the training set and saved a new CSV as processed_train.csv

The Variables

train <- read.csv("data/processed/processed_train.csv")
names(train)

##  [1] "EventId"                     "DER_mass_MMC"               
##  [3] "DER_mass_transverse_met_lep" "DER_mass_vis"               
##  [5] "DER_pt_h"                    "DER_deltaeta_jet_jet"       
##  [7] "DER_mass_jet_jet"            "DER_prodeta_jet_jet"        
##  [9] "DER_deltar_tau_lep"          "DER_pt_tot"                 
## [11] "DER_sum_pt"                  "DER_pt_ratio_lep_tau"       
## [13] "DER_met_phi_centrality"      "DER_lep_eta_centrality"     
## [15] "PRI_tau_pt"                  "PRI_tau_eta"                
## [17] "PRI_tau_phi"                 "PRI_lep_pt"                 
## [19] "PRI_lep_eta"                 "PRI_lep_phi"                
## [21] "PRI_met"                     "PRI_met_phi"                
## [23] "PRI_met_sumet"               "PRI_jet_num"                
## [25] "PRI_jet_leading_pt"          "PRI_jet_leading_eta"        
## [27] "PRI_jet_leading_phi"         "PRI_jet_subleading_pt"      
## [29] "PRI_jet_subleading_eta"      "PRI_jet_subleading_phi"     
## [31] "PRI_jet_all_pt"              "Label"

signal <- which(train$Label == "s")

require(ggplot2)

## Loading required package: ggplot2

CreatePlot <- function(data, v, yintercept = c()) {
    p <- ggplot(data, aes(x = EventId, y = data[, v], color = Label), environment = environment())
    l <- p + labs(y = v)
    if (length(yintercept) == 0) {
        print(l + geom_point())
    } else {
        print(l + geom_point() + geom_hline(yintercept = yintercept))
    }
}

Primitive Variables

PRI_tau_pt

The transverse momentum \( \sqrt{p^2_x+p^2_y} \), or the momentum perpendicular to the beam line, of the hadronic tau measured in GeV

CreatePlot(train, "PRI_tau_pt")

plot of chunk pri_tau_pt_fig 

ptp <- train$PRI_tau_pt
summary(ptp)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    20.0    24.6    31.8    38.7    45.0   764.0

Most observations are < 175 GeV, let's replot on that subset

ptp.175 <- subset(train, PRI_tau_pt < 175)
CreatePlot(ptp.175, "PRI_tau_pt")

plot of chunk subset_ptp_175

It looks like the slower the transverse momentum, the more likely the observation is background noise. Let's replot for < 50

ptp.50 <- subset(train, PRI_tau_pt < 50)
CreatePlot(ptp.50, "PRI_tau_pt")

plot of chunk subset_ptp_50

PRI_tau_eta

The pseudorapidity η (the angle from the beam axis) of the hadronic tau.

CreatePlot(train, "PRI_tau_eta")

plot of chunk pri_tau_eta_fig

It looks like a higher quantity of signal observations exist within -1.5 to 1.5 than outside this range.

signal.obs <- subset(train, Label == "s", select = PRI_tau_eta)
# quantity signal in range -1.5 to 1.5
in.range <- subset(signal.obs, PRI_tau_eta >= -1.5 & PRI_tau_eta <= 1.5)
nrow(in.range)/nrow(signal.obs)

## [1] 0.7974

Almost 80% of signal observations fall in the -1.5 to 1.5 range. Is it also more likely that an observation outside of this range is background?

outside.range <- subset(train, PRI_tau_eta < -1.5 | PRI_tau_eta > 1.5, select = Label)
table(outside.range)

## outside.range
##     b     s 
## 49179 17355

length(which(outside.range == "b"))/nrow(outside.range)

## [1] 0.7392

About 74% of observations outside the range -1.5 to 1.5 are background noise.

PRI_tau_phi

The azimuthal angle φ, from -pi to pi in the spherical coordinate system, of the hadronic tau around the beam axis.

CreatePlot(train, "PRI_tau_phi")

plot of chunk pri_tau_phi_fig 

# all observations
summary(train$PRI_tau_phi)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -3.1400 -1.5800 -0.0330 -0.0082  1.5600  3.1400

# just the signal
summary(train[signal, "PRI_tau_phi"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -3.1400 -1.5800 -0.0590 -0.0192  1.5400  3.1400

# just the background
summary(train[-signal, "PRI_tau_phi"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -3.1400 -1.5700 -0.0170 -0.0024  1.5800  3.1400

PRI_lep_pt

The transverse momentum \( \sqrt{p^2_x+p^2_y} \), or the momentum perpendicular to the beam line, of the lepton (electron or muon) measured in GeV

CreatePlot(train, "PRI_lep_pt")

plot of chunk pri_lep_pt_fig

Replotting to < 200

plp.200 <- subset(train, PRI_lep_pt < 200)
CreatePlot(plp.200, "PRI_lep_pt")

plot of chunk pri_lep_pt_200 

summary(train$PRI_lep_pt)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    26.0    32.4    40.5    46.7    53.4   560.0

PRI_lep_eta

The pseudorapidity η (the angle from the beam axis) of the lepton.

CreatePlot(train, "PRI_lep_eta")

plot of chunk pri_lep_eta_fig

Just as with the pseudorapidity of the hadronic tau it appears the signal clusters in the range of -1.5 to 1.5

signal.obs <- subset(train, Label == "s", select = PRI_lep_eta)
# quantity signal in range -1.5 to 1.5
in.range <- subset(signal.obs, PRI_lep_eta >= -1.5 & PRI_lep_eta <= 1.5)
nrow(in.range)/nrow(signal.obs)

## [1] 0.7837

Aabout 78% of signal observations fall in the -1.5 to 1.5 range. 

outside.range <- subset(train, PRI_lep_eta < -1.5 | PRI_lep_eta > 1.5, select = Label)
table(outside.range)

## outside.range
##     b     s 
## 56096 18534

length(which(outside.range == "b"))/nrow(outside.range)

## [1] 0.7517

About 75% of observations outside the range -1.5 to 1.5 are background noise.

PRI_lep_phi

The azimuthal angle φ, from -pi to pi in the spherical coordinate system, of the lepton around the beam axis.

CreatePlot(train, "PRI_lep_phi")

plot of chunk pri_lep_phi_fig 

# all observations
summary(train$PRI_lep_phi)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -3.1400 -1.5200  0.0860  0.0435  1.6200  3.1400

# just the signal
summary(train[signal, "PRI_lep_phi"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -3.1400 -1.5000  0.1030  0.0539  1.6300  3.1400

# just the background
summary(train[-signal, "PRI_lep_phi"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -3.1400 -1.5300  0.0760  0.0381  1.6100  3.1400

PRI_met The missing transverse energy \( \overrightarrow{E}^{miss}_T \) 

CreatePlot(train, "PRI_met")

plot of chunk pri_met 

summary(train$PRI_met)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     0.1    21.4    34.8    41.7    51.9  2840.0

# just the signal
summary(train[signal, "PRI_met"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     0.2    17.9    31.9    42.7    54.5  2840.0

# just the background
summary(train[-signal, "PRI_met"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     0.1    23.3    35.9    41.2    51.0   951.0

That one outlier really makes this blurry…

pm.500 <- subset(train, PRI_met < 500)
CreatePlot(pm.500, "PRI_met")

plot of chunk pri_met_500 

pm.200 <- subset(train, PRI_met < 200)
CreatePlot(pm.200, "PRI_met")

plot of chunk pri_met_200 

pm.100 <- subset(train, PRI_met < 100)
CreatePlot(pm.100, "PRI_met")

plot of chunk pri_met_100

There appears to be a lot of signal clustered < 25

under.25 <- subset(train[signal, ], PRI_met < 25)
nrow(under.25)/length(signal)

## [1] 0.3842

About 38% of the signal is < 25

PRI_met_phi

The azimuth angle φ of the missing transverse energy.

CreatePlot(train, "PRI_met_phi")

plot of chunk pri_met_phi 

summary(train$PRI_met_phi)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -3.1400 -1.5800 -0.0240 -0.0101  1.5600  3.1400

# just the signal
summary(train[signal, "PRI_met_phi"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -3.1400 -1.5500  0.0190  0.0086  1.5800  3.1400

# just the background
summary(train[-signal, "PRI_met_phi"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -3.1400 -1.5800 -0.0440 -0.0199  1.5500  3.1400

PRI_met_sumet

The total transverse energy in the detector.

CreatePlot(train, "PRI_met_sumet")

plot of chunk pri_met_sumet 

summary(train$PRI_met_sumet)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    13.7   123.0   180.0   210.0   263.0  2000.0

# just the signal
summary(train[signal, "PRI_met_sumet"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    13.7   146.0   208.0   234.0   291.0  1390.0

# just the background
summary(train[-signal, "PRI_met_sumet"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    18.5   114.0   166.0   197.0   246.0  2000.0

pms.750 <- subset(train, PRI_met_sumet < 750)
CreatePlot(pms.750, "PRI_met_sumet")

plot of chunk pri_met_sumet_750 

pms.300 <- subset(train, PRI_met_sumet < 300)
CreatePlot(pms.300, "PRI_met_sumet")

plot of chunk pri_met_sumet_300 

pms.200 <- subset(train, PRI_met_sumet < 200)
CreatePlot(pms.200, "PRI_met_sumet")

plot of chunk pri_met_sumet_200

PRI_jet_num

The number of jets (a narrow cone of hadrons and other particles produced by the hadronization of a quark or gluon) http://en.wikipedia.org/wiki/Jet_(particle_physics)

ggplot(data = train, aes(x = PRI_jet_num, fill = Label)) + geom_bar(position = position_dodge())

## stat_bin: binwidth defaulted to range/30. Use 'binwidth = x' to adjust this.

plot of chunk pri_jet_num

PRI_jet_leading_pt The transverse momentum \( \sqrt{p^2_x+p^2_y} \) of the leading jet, that is the jet with largest transverse momentum (undefined if PRI jet num = 0).

CreatePlot(train, "PRI_jet_leading_pt")

plot of chunk pri_jet_leading_pt 

pjlp.na <- subset(train, PRI_jet_leading_pt == -999, select = Label)
table(pjlp.na)

## pjlp.na
##     b     s 
## 74421 25492

length(which(pjlp.na == "b"))/nrow(pjlp.na)

## [1] 0.7449

75% of undefined are background.

pjlp <- subset(train, PRI_jet_leading_pt != -999)
CreatePlot(pjlp, "PRI_jet_leading_pt")

plot of chunk pri_jet_leading_pt_defined 

summary(pjlp$PRI_jet_leading_pt)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    30.0    44.4    65.6    84.8   103.0  1120.0

pjlp.signal <- which(pjlp$Label == "s")
# just the signal
summary(pjlp[pjlp.signal, "PRI_jet_leading_pt"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    30.0    50.1    74.4    92.9   114.0   738.0

# just the background
summary(pjlp[-pjlp.signal, "PRI_jet_leading_pt"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    30.0    41.7    60.1    79.4    95.2  1120.0

PRI_jet_leading_eta

The pseudorapidity η of the leading jet (undefined if PRI jet num = 0).

CreatePlot(train, "PRI_jet_leading_eta")

plot of chunk pri_jet_leading_eta 

pjle.na <- subset(train, PRI_jet_leading_eta == -999, select = Label)
table(pjle.na)

## pjle.na
##     b     s 
## 74421 25492

length(which(pjle.na == "b"))/nrow(pjle.na)

## [1] 0.7449

74% of undefined are background.

pjle <- subset(train, PRI_jet_leading_eta != -999)
CreatePlot(pjle, "PRI_jet_leading_eta", c(-1.5, 1.5))

plot of chunk pri_jet_leading_eta_defined 

pjle.inrange <- which(pjle$PRI_jet_leading_eta >= -1.5 & pjle$PRI_jet_leading_eta <= 
    1.5)
table(pjle[pjle.inrange, "Label"])

## 
##     b     s 
## 55373 27680

nrow(subset(pjle[pjle.inrange, ], Label == "b"))/length(pjle.inrange)

## [1] 0.6667

67% of defined observations in the -1.5 to 1.5 range are background

What percentange of defined observations outside the -1.5 to 1.5 range are signal?

out.range <- subset(pjle, PRI_jet_leading_eta < -1.5 | PRI_jet_leading_eta > 
    1.5, select = "Label")
table(out.range)

## out.range
##     b     s 
## 34539 32495

summary(pjle$PRI_jet_leading_eta)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  -4.500  -1.340   0.000  -0.003   1.340   4.500

pjle.signal <- which(pjle$Label == "s")
# just the signal
summary(pjle[pjle.signal, "PRI_jet_leading_eta"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  -4.500  -1.630  -0.005  -0.003   1.620   4.500

# just the background
summary(pjle[-pjle.signal, "PRI_jet_leading_eta"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  -4.500  -1.170   0.002  -0.004   1.170   4.490

PRI_jet_leading_phi

The azimuth angle φ of the leading jet (undefined if PRI jet num = 0).

CreatePlot(train, "PRI_jet_leading_phi")

plot of chunk pri_jet_leading_phi 

pjlphi.na <- subset(train, PRI_jet_leading_phi == -999, select = Label)
table(pjlphi.na)

## pjlphi.na
##     b     s 
## 74421 25492

length(which(pjlphi.na == "b"))/nrow(pjlphi.na)

## [1] 0.7449

75% of undefined are background

pjlphi <- subset(train, PRI_jet_leading_phi != -999)
CreatePlot(pjlphi, "PRI_jet_leading_phi")

plot of chunk pri_jet_leading_phi_defined 

summary(pjlphi$PRI_jet_leading_phi)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -3.1400 -1.5800 -0.0330 -0.0124  1.5600  3.1400

pjlphi.signal <- which(pjlphi$Label == "s")
# just the signal
summary(pjlphi[pjlphi.signal, "PRI_jet_leading_phi"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -3.1400 -1.5900 -0.0360 -0.0129  1.5700  3.1400

# just the background
summary(pjlphi[-pjlphi.signal, "PRI_jet_leading_phi"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -3.1400 -1.5800 -0.0310 -0.0121  1.5600  3.1400

PRI_jet_subleading_pt

The transverse momentum \( \sqrt{p^2_x+p^2_y} \) of the second leading jet. (undefined if PRI jet num ≤ 1).

CreatePlot(train, "PRI_jet_subleading_pt")

plot of chunk pri_jet_subleading_pt 

pjsp.na <- subset(train, PRI_jet_subleading_pt == -999, select = Label)
table(pjsp.na)

## pjsp.na
##      b      s 
## 124255  53202

length(which(pjsp.na == "b"))/nrow(pjsp.na)

## [1] 0.7002

70% of undefined are background

pjsp <- subset(train, PRI_jet_subleading_pt != -999)
CreatePlot(pjsp, "PRI_jet_subleading_pt")

plot of chunk pri_jet_subleading_pt_defined 

summary(pjsp$PRI_jet_subleading_pt)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    30.0    37.3    47.9    57.7    66.6   721.0

pjsp.signal <- which(pjsp$Label == "s")
# just the signal
summary(pjsp[pjsp.signal, "PRI_jet_subleading_pt"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    30.0    37.8    48.6    56.9    66.8   422.0

# just the background
summary(pjsp[-pjsp.signal, "PRI_jet_subleading_pt"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    30.0    37.0    47.4    58.3    66.5   721.0

PRI_jet_subleading_phi

The azimuth angle φ of the subleading jet (undefined if PRI jet num ≤ 1).

CreatePlot(train, "PRI_jet_subleading_phi")

plot of chunk pri_jet_subleading_phi 

pjsphi.na <- subset(train, PRI_jet_subleading_phi == -999, select = Label)
table(pjsphi.na)

## pjsphi.na
##      b      s 
## 124255  53202

length(which(pjsphi.na == "b"))/nrow(pjsphi.na)

## [1] 0.7002

70% of undefined are background

pjsphi <- subset(train, PRI_jet_subleading_phi != -999)
CreatePlot(pjsphi, "PRI_jet_subleading_phi")

plot of chunk pri_jet_subleading_phi_defined 

summary(pjsphi$PRI_jet_subleading_phi)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -3.1400 -1.5800 -0.0020 -0.0016  1.5800  3.1400

pjsphi.signal <- which(pjsphi$Label == "s")
# just the signal
summary(pjsphi[pjsphi.signal, "PRI_jet_subleading_phi"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -3.1400 -1.5900 -0.0240 -0.0143  1.5600  3.1400

# just the background
summary(pjsphi[-pjsphi.signal, "PRI_jet_subleading_phi"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -3.1400 -1.5700  0.0160  0.0087  1.5900  3.1400

PRI_jet_all_pt

The scalar sum of the transverse momentum of all the jets.

CreatePlot(train, "PRI_jet_all_pt")

plot of chunk pri_jet_all_pt 

pjap.500 <- subset(train, PRI_jet_all_pt < 500)
CreatePlot(pjap.500, "PRI_jet_all_pt")

plot of chunk pri_jet_all_pt_500 

pjap.300 <- subset(train, PRI_jet_all_pt < 300)
CreatePlot(pjap.300, "PRI_jet_all_pt")

plot of chunk pri_jet_all_pt_300 

summary(train$PRI_jet_all_pt)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     0.0     0.0    40.5    73.1   110.0  1630.0

# just the signal
summary(train[signal, "PRI_jet_all_pt"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     0.0     0.0    68.0    91.3   139.0  1190.0

# just the background
summary(train[-signal, "PRI_jet_all_pt"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     0.0     0.0    33.6    63.6    89.8  1630.0

Derived Variables

DER_mass_MMC

The estimated mass of the Higgs boson candidate. May be undefined.

CreatePlot(train, "DER_mass_MMC")

plot of chunk der_mass_mmc 

dmm.na <- subset(train, DER_mass_MMC == -999, select = Label)
table(dmm.na)

## dmm.na
##     b     s 
## 35279  2835

length(which(dmm.na == "b"))/nrow(dmm.na)

## [1] 0.9256

93% of undefined are background

dmm <- subset(train, DER_mass_MMC != -999)
CreatePlot(dmm, "DER_mass_MMC")

plot of chunk der_mass_mmc_defined 

dmm.250 <- subset(dmm, DER_mass_MMC < 250)
CreatePlot(dmm.250, "DER_mass_MMC", c(75, 175))

plot of chunk der_mass_mmc_250 

dmm.signal <- subset(dmm, Label == "s")
dmm.signal.inrange <- subset(dmm.signal, DER_mass_MMC >= 75 & DER_mass_MMC <= 
    175)
nrow(dmm.signal.inrange)/nrow(dmm.signal)

## [1] 0.9702

97% of defined signal observations lie in range 75 to 175

DER_mass_transverse_met_lep

The transverse mass between the missing transverse energy and the lepton.

CreatePlot(train, "DER_mass_transverse_met_lep")

plot of chunk der_mass_transverse_met_lep 

dmtml.200 <- subset(train, DER_mass_transverse_met_lep < 200)
CreatePlot(dmtml.200, "DER_mass_transverse_met_lep")

plot of chunk der_mass_transverse_met_lep_200 

dmtml.150 <- subset(train, DER_mass_transverse_met_lep < 150)
CreatePlot(dmtml.150, "DER_mass_transverse_met_lep", 75)

plot of chunk der_mass_transverse_met_lep_150 

dmtml.signal.75 <- subset(train[signal, ], DER_mass_transverse_met_lep < 75)
nrow(dmtml.signal.75)/nrow(train[signal, ])

## [1] 0.9257

93% of all signal observations are < 75

summary(train$DER_mass_transverse_met_lep)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     0.0    19.2    46.5    49.2    73.6   690.0

# just the signal
summary(train[signal, "DER_mass_transverse_met_lep"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     0.0    10.4    24.4    32.0    45.7   570.0

# just the background
summary(train[-signal, "DER_mass_transverse_met_lep"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     0.0    31.4    60.8    58.2    80.2   690.0

DER_mass_vis

The total invariant mass (total energy and momentum) of the hadronic tau and the lepton.

CreatePlot(train, "DER_mass_vis", 150)

plot of chunk der_mass_vis 

dmv.150 <- subset(train, DER_mass_vis < 150)
CreatePlot(dmv.150, "DER_mass_vis")

plot of chunk der_mass_vis_150 

summary(train$DER_mass_vis)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     6.3    59.4    73.8    81.2    92.3  1350.0

# just the signal
summary(train[signal, "DER_mass_vis"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     7.5    67.4    79.4    80.4    91.9   790.0

# just the background
summary(train[-signal, "DER_mass_vis"])

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     6.3    55.5    69.7    81.6    92.7  1350.0